Stationary random subgroups and injectivity radius of hyperbolic manifolds

There is a long tradition of using probabilistic methods to solve geometric problems. I will present one such result. Namely, I will show that if the bottom of the spectrum of the Laplacian on a hyperbolic n manifold M is equal to that of its universal cover (or equivalently the fundamental group has exponential growth rate at most (n-1)/2) then M has points with arbitrary large injectivity radius.